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23Chris Parkes  Solve Schrödinger for the eigenstates of H :  of the form  with complex parameters p and q satisfying  Time evolution of the eigenstates: Time evolution of neutral mesons mixed states (2/4) Compare with K s, K L as mixtures of K 0, K 0 If equal mixtures, like K 1 K 2

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25Chris Parkes  Evolution of weak/flavour eigenstates:  Time evolution of mixing probabilities: Interference term Time evolution of neutral mesons mixed states (4/4) i.e. if start with P 0, what is probability that after time t that have state P 0 ? decay terms Parameter x determines “speed” of oscillations compared to the lifetime

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26Chris Parkes Hints: for proving probabilities Starting point Turn this around, gives Time evolution Use these to find

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30Chris Parkes  In 1963 N. Cabibbo made the first step to formally incorporate strangeness violation in weak decays 1) For the leptons, transitions only occur within a generation 2) For quarks the amount of strangeness violation can be neatly described in terms of a rotation, where  c =13.1 o Cabibbo rotation and angle (1/3) Weak force transitions u d’ = dcos c + ssin c W+W+ Idea: weak interaction couples to different eigenstates than strong interaction weak eigenstates can be written as rotation of strong eigenstates

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34Chris Parkes The GIM mechanism (2/2)  There is also an interesting symmetry between quark generations: u d’=cos( c )d+sin( c )s W+W+ c s’=-sin( c )d+cos( c )s W+W+ Cabibbo mixing matrix The d quark as seen by the W, the weak eigenstate d’, is not the same as the mass eigenstate (the d)

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40Chris Parkes  As the CKM matrix elements are connected to probabilities of transition, the matrix has to be unitary: CKM matrix – number of parameters (1/2) Values of elements: a purely experimental matter In general, for N generations, N 2 constraints Sum of probabilities must add to 1 e.g. t must decay to either b, s, or d so  Freedom to change phase of quark fields 2N-1 phases are irrelevant (choose i and j, i≠j)  Rotation matrix has N(N-1)/2 angles

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49 CKM – Unitarity Triangle Three complex numbers, which sum to zero Divide by so that the middle element is 1 (and real) Plot as vectors on an Argand diagram If all numbers real – triangle has no area – No CP violation Real Imaginary Hence, get a triangle ‘Unitarity’ or ‘CKM triangle’ Triangle if SM is correct. Otherwise triangle will not close, Angles won’t add to 180 o

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51Chris Parkes The Unitarity Triangle(s) & the , ,  angles  Area of all the triangles is the same ( 6 A 2  )  Jarlskog invariant J, related to how much CP violation  Two triangles (db) and (ut) have sides of similar size Easier to measure, (db) is often called THE unitarity triangle

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52Chris Parkes CKM Triangle - Experiment Find particle decays that are sensitive to measuring the angles (phase difference) and sides (probabilities) of the triangles Measurements constrain the apex of the triangle Measurements are consistent We will discuss how to experimentally measure the sides / angles CKM model works, 2008 Nobel prize

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53Chris Parkes Key Points So Far K 0, K 0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states CP Violated (a tiny bit) in Kaon decays Describe this through K s, K L as mixture of K 0 K 0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P 0 P 0 Mass differences and width determine the rates of oscillations Very different for different mesons (B s,B,D,K) Weak and mass eigenstates of quarks are not the same Describe through rotation matrix – Cabibbo (2 generations), CKM (3 generations) CP Violation included by making CKM matrix elements complex Depict matrix elements and their relationships graphically with CKM triangle

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Types of CP violation We discussed earlier how CP violation can occur in Kaon (or any P 0 ) mixing if p≠q. We didn’t consider the decay of the particle – this leads to two more ways to violate CP

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56Chris Parkes Occurs when a decay and its CP-conjugate decay have a different probability Decay amplitudes can be written as: Two types of phase:  Strong phase: CP conserving, contribution from intermediate states  Weak phase  : complex phase due to weak interactions 1) CP violation in decay (also called direct CP violation) Valid for both charged and neutral particles P (other types are neutral only since involve oscillations)

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58Chris Parkes  Say we have a particle* such that P 0  f and P 0  f are both possible  There are then 2 possible decay chains, with or without mixing!  Interference term depends on  Can put and get but * Not necessary to be CP eigenstate 3) CP violation in the interference of mixing and decay CP can be conserved in mixing and in decay, and still be violated overall !

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59Chris Parkes Key Points So Far K 0, K 0 are not CP eigenstates – need to make linear combination Short lived and long-lived Kaon states CP Violated (a tiny bit) in Kaon decays Describe this through K s, K L as mixture of K 0 K 0 Neutral mesons oscillate from particle to anti-particle Can describe neutral meson oscillations through mixture of P 0 P 0 Mass differences and width determine the rates of oscillations Very different for different mesons (B s,B,D,K) Weak and mass eigenstates of quarks are not the same Describe through rotation matrix – Cabibbo (2 generations), CKM (3 generations) CP Violation included by making CKM matrix elements complex Depict matrix elements and their relationships graphically with CKM triangle Three ways for CP violation to occur Decay Mixing Interference between decay and mixing